2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Differential Equations, Conference 08, 2002, pp 133–155. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Singular p-harmonic functions and related quasilinear equations on manifolds ^{∗}

Laurent Véron

Abstract We give here an overview of some recent developments in the study of

the description of singular solutions of

−∇.(|∇u|^{p−2}∇u) + ε|u|^{q−1}u = 0

in ℝ^{N} \ {0}, where p > 1, ε ∈ {0, 1, −1} and q ≥ p − 1.

1 Introduction

Let Ω be a domain in ℝ^{N} containing 0, N ≥ 2, and let

A : Ω × ℝ × ℝ^{N} ↦ ℝ^{N}, and B : Ω × ℝ × ℝ^{N} ↦ ℝ,

be two Caratheodory functions. Then a classical problem is the study of the behaviour near 0 of a solution u of

−∇.A(x, u, ∇u) + B(x, u, ∇u) = 0  (1.1)

in Ω^{∗} = Ω \ {0}. Besides the well known linear case, the first striking results in the nonlinear case were obtained by Serrin in 1964 in a series of celebrated articles [11, 12]. Under the assumptions

(i) A(x, r, Q).Q ≥ c_{1}|Q|^{p}
(ii) |A(x, r, Q)| ≤ c_{2}|Q|^{p−1} + c_{3}  (1.2)
(iii) |B(x, r, Q)| ≤ c_{4}|Q|^{p−1} + c_{5}|r|^{p−1} + c_{6}

for any (x, r, Q) ∈ Ω × ℝ × ℝ^{N} ↦ ℝ^{N}, where the c_{i} are positive constants and N ≥ p > 1. Serrin’s results assert that any nonnegative weak solution u of (1.1) in Ω^{∗} belonging to W_{loc}^{1,p}(Ω^{∗}) is either extendable by continuity as a C(Ω) ∩ W_{loc}^{1,p}(Ω)-solution of the same equation in whole Ω, or satisfies

θ ≤ \frac{u(x)}{μ_{p}(x)} ≤ θ^{−1},  (1.3)

∗Mathematics Subject Classifications: 35J50, 35J60.

Key words: p-harmonic, singularity, degenerate equations.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published October 21, 2002.

near 0, for some positive θ, in which formula the functions μ_{p} are defined in ℝ^{N} \ {0} by

μ_{p}(x) = \begin{cases}|x|^{(p−N)/(p−1)} & if 1 < p < N, \\ ln(1/|x|) & if p = N. \end{cases}  (1.4)

A series of extensions were obtained in the eighties in the case

A(x, r, Q) = |Q|^{p−2}Q,

where the diffusion operator ∇.A(x, u, ∇u) is called the p-Laplace: by Kichenassamy and Véron [9] in the case B(x, r, Q) ≡ 0; Vazquez and Véron [17], Friedman and Véron [5] in the case B(x, r, Q) = |r|^{q−1}r with q > p−1; Guedda and Véron [7], Bidaut-Véron [1], Serrin and Zou [13] in the case B(x, r, Q) = −|r|^{q−1}r, always in assuming q > p − 1. We shall present below an overview or the results of these different authors, writing the equation (1.1) in the form

−∇.(|∇u|^{p−2}∇u) + ε|u|^{q−1}u = 0,  (1.5)

with ε = 1, −1 or 0. We put emphasis on separable solutions that are solutions of the form

u(r, σ) = r^{−β}ω(σ), (r, σ) ∈ (0, ∞) × S^{N−1}.

Thus β = β_{q} = p/(q + 1 − p) and the relation

−∇_{σ}. ((ω^{2} + |∇_{σ}ω|^{2})^{p/2−1}∇_{σ}ω) + ε|ω|^{q−1}ω
= β_{q}((β_{q} + 1)(p − 1) + 1 − N)(ω^{2} + |∇_{σ}ω|^{2})^{p/2−1}ω,

holds on S^{N−1}. This equation is not the usual Euler equation of a functional, which makes it more difficult study. However, we give a few results of existence and uniqueness of solutions.

2 Singular p-harmonic functions

By looking for radial solutions of the p-Laplace equation

−∇.(|∇u|^{p−2}∇u) = 0,  (2.1)

in ℝ^{N} \ {(0)}, we find that the only solutions are the functions

u = C_{1}μ_{p} + C_{2}

where the C_{i} are arbitrary constants.The first result obtained by Kichenassamy and Véron in [9] pointed out that any nonnegative singular p-hamonic functions is asymptotically radial near its singularities. They proved the following result.

Theorem 2.1 Assume 1 < p ≤ N and u ∈ W_{loc}^{1,p}(Ω^{∗}) is nonnegative and satisfies (2.1) in Ω^{∗}. Then there exists γ ∈ ℝ_{+} such that

u − γμ_{p} ∈ L_{loc}^{∞}(Ω).  (2.2)

Moreover

lim_{x→0} |x|^{(N−1)/(p−1)}∇(u − γμ_{p})(x) = 0,  (2.3)

and the following equation holds in the sense of distributions in Ω

−∇.(|∇u|^{p−2}∇u) = c_{N,p}γ^{p−1}δ_{0},  (2.4)

for some positive constant c_{N,p}.

The proof is based on the a priori estimate

u(x) ≤ Cμ_{p}(x)

for 0 < |x| ≤ R, for some C > 0 and R > 0 (this follows from Serrin’s result), the scaling transformation

T_{r}(u)(ξ) = u(rξ)/μ(r)

and a version of the strong maximum principle which was first noticed by Tolksdorff [14]. Actually, the positivity assumption can be relaxed and replaced by

u/μ_{p} ∈ L^{∞}(B_{R}),  (2.5)

since Serrin’s result asserts that any nonnegative singular p-harmonic function does satisfy this estimate. As a consequence, existence and uniqueness of a solution to the singular Dirichlet problem

\begin{matrix}−∇.(|∇u|^{p−2}∇u) = c_{N,p}|γ|^{p−2}γδ_{0}, in D′(Ω), \\ u = g, on ∂Ω, \end{matrix}  (2.6)

can be proved.

Corollary 2.2 Assume 1 < p ≤ N, Ω is bounded with a C^{2} boundary, g ∈ L^{∞}(∂Ω) ∩ W^{1−1/p,p}(∂Ω) and γ ∈ ℝ. Then there exists a unique u ∈ C^{1}(Ω^{∗}) such that |∇u|^{p−1} ∈ L^{1}(Ω) satisfying (2.6) and (2.5). Moreover (2.2) and (2.3) hold.

Another consequence is the following singular Liouville type result.

Corollary 2.3 Assume 1 < p ≤ N, and u ∈ C^{1}(ℝ^{N} \ {0}) is p-harmonic in ℝ^{N} \ {0} and satisfies |u(x)| ≤ a|μ_{p}(x)| + b, for some positive constants a and b. Then there exist two real numbers α and β such that

u = αμ_{p} + β.

If we look for singular p-harmonic functions u in ℝ^{N} \ {0} under the form

u(x) = |x|^{−β}ω(x/|x|) = r^{−β}ω(σ),  (2.7)

where (r, σ) ∈ (0, ∞) × S^{N−1} are the spherical coordinates, then

−∇_{σ}.((β^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}∇_{σ}ω) = λ(β^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}ω, (2.8)

where ∇_{σ}. is the divergence operator acting on C^{1} vector fields on the unit (N-1)-sphere S^{N−1} and ∇_{σ} is the tangential gradient, identified with the covariant derivative on S^{N−1} for the Riemannian structure induced by the imbedding of S^{N−1} into ℝ^{N}, and

λ = β((β + 1)(p − 1) + 1 − N).

When N = 2 and ω(x/|x|) = ω(φ) is a 2π- periodic function, equation (2.7) becomes

((β^{2}ω^{2} + ω_{φ}^{2})^{(p−2)/2}ω_{φ})_{φ} + ((β + 1)(p − 1) − 1)β(β^{2}ω^{2} + ω_{φ}^{2})^{(p−2)/2}ω = 0. (2.9)

Putting Y = ω_{φ}/ω, and β_{0} = (2 − p)/(p − 1) yields to

(\frac{β}{Y^{2} + β^{2}} − \frac{β + 1}{Y^{2} + β(β − β_{0})})Y_{φ} = 1.

This equation is completely integrable [9], and the following result is proved.

Theorem 2.4 Assume p > 1, then for each positive integer k there exist a β_{k} and ω_{k} : ℝ ↦ ℝ with least period 2π/k, of class C^{∞} such that

u(x) = |x|^{−β_{k}}ω_{k}(x/|x|),  (2.10)

is p-harmonic in ℝ^{2} \ {0}; β_{k} is the positive root of

(β + 1)^{2} = (1 + 1/k)^{2} (β^{2} + β(p − 2)/(p − 1)) .  (2.11)

The couple (β_{k}, ω_{k}) is unique, up to translation and homothety over ω_{k}.

In the case of regular p-harmonic functions in the plane, which means that the exponent β = −β̃ in (2.7) is negative, the stationary equation becomes

((β̃^{2}ω̃^{2} +ω̃_{φ}^{2})^{(p−2)/2}ω̃_{φ})_{φ} +((β̃−1)(p−1)−1)β̃(β̃^{2}ω̃^{2} +ω̃_{φ}^{2})^{(p−2)/2}ω̃ = 0. (2.12)

Kroll and Mazja [8] obtained the complete set of solutions of (2.12):

Theorem 2.5 For each positive integer k there exists a couple (β̃_{k}, ω̃_{k}), unique up to translation and homothety over ω̃_{k} such that

x ↦ u(x) = |x|β̃_{k} ω̃_{k}(x/|x|),  (2.13)

is p-harmonic in ℝ^{2}. The exponent β̃_{k} is the root larger than 1 of the algebraic equation

(β̃ − 1)^{2} = (1 − 1/k)^{2} (β̃^{2} − β̃(p − 2)/(p − 1)) .  (2.14)

The derivation of regular or singular p-harmonic functions follows in higher dimension under a splitted form. For example, if N = 3 with (x_{1}, x_{2}, x_{3}) the canonical coordinates in ℝ^{3}, we put

x_{1} = r cos φ sin θ, x_{2} = r sin φ sin θ, x_{3} = r cos θ,

where r > 0, φ ∈ [0, 2π], θ ∈ [0, π]. Equation (2.8) takes the form

−\frac{∂}{∂θ} (sin θ (β^{2}ω^{2} + ω_{θ}^{2} + sin^{−2} θ ω_{φ}^{2})^{(p−2)/2} ω_{θ})
−\frac{∂}{∂φ} (sin^{−1} θ (β^{2}ω^{2} + ω_{θ}^{2} + sin^{−2} θ ω_{φ}^{2})^{(p−2)/2} ω_{φ})
=β(β(p − 1) + p − 3) sin θ (β^{2}ω^{2} + ω_{θ}^{2} + sin^{−2} θ ω_{φ}^{2})^{(p−2)/2} ω.

(2.15)

We set

ω(φ, θ) = sin^{−β} θ v(φ) = sinβ̃ θ v(φ),

then v satisfies (2.12). Thanks to Theorem 2.5 the set of singular (resp. regular) p-harmonic functions under the form

u(r, φ, θ) = r^{−β} sin^{−β} θ v(φ),

resp.

u(r, φ, θ) = rβ̃ sinβ̃ θ v(φ),

is explicitly known. Another way for constructing non-isotropic singular pharmonic functions is to use Tolksdorf’s shooting method [14].

Theorem 2.6 Let S ⊂ S^{N−1} be a connected and open, with a C^{2} relative boundary ∂S. Then there exist a unique couple (β, ω), with β > 0, ω ∈ C^{1}(S), ω > 0 in S, vanishing on ∂S, with maximal value 1 such that the function u defined by (2.7) is p-harmonic in ℝ^{N} \ {0}.

Proof Put K_{S}(R, R′) = {(r, σ) : σ ∈ S, R < r < R′} and B_{S}(R, R′) = {(r, σ) : σ ∈ ∂S, R < r < R′}. Let g be defined by

g(x) = \begin{cases}2 − |x| & if |x| ≤ 2, \\ 0 & if |x| ≥ 2. \end{cases}

For n ≥ 2 we denote by u_{n} the unique solution of

\begin{matrix}−∇.(|∇u_{n}|^{p−2}∇u_{n}) = 0 in K_{S}(1, n), \\ u_{n} = g on B_{S}(1, n). \end{matrix}

Since Hopf maximum principle holds [14], u_{n} is positive in K_{S}(1, n). The sequence \{u_{n}\} is increasing and locally bounded in the C_{loc}^{1,α} topology of \overline{K_{S}(1, ∞)}.

Thus it converges in C_{loc}^{1}(\overline{K_{S}(1, ∞}) to some u which is positive and satisfies

\begin{matrix}−∇.(|∇u|^{p−2}∇u) = 0 in K_{S}(1, ∞), \\ u = g on B_{S}(1, ∞), \\ lim_{|x|→∞} u(x) = 0. \end{matrix}  (2.16)

The function

R ↦ C(R) = sup_{x∈K_{S}(1,∞)} u(x)

is decreasing and the supremum is achieved for |x| = R. One of the key idea is called the equivalence principle [14, Lemma 2.1], Lemma 2.1, which asserts that

u(Rx) ≤ (1 − ε(R − 1))u(x),  (2.17)

for some ϵ > 0 and any R ∈ (1, 2). Thus there exists k > 0 such that C(R) ≤ kC(2R) for any R ≥ 3. Then

|∇u(x)| ≤ C(|x|)|x|^{−1}, and |∇u(x) − ∇u(x′)| ≤ C(|x|)|x|^{−1−α}|x − x′|^{α},

for some C > 0 and 1 ≤ |x| ≤ |x′|. Putting

u_{R}(x) = u(Rx)/C(R),

it follows that for any compact subset K of \overline{K_{S}(0, ∞)}\\{0\} there exists C(K) > 0 such that

∥u_{R}∥_{C^{1,α}(K)} ≤ C(K).

Thus there exist a sequence R_{n} → ∞ and a p-harmonic function u^{∗} in K_{S}(0, ∞) such that u_{R_{n}} → u^{∗} in the C_{loc}^{1} topology of \overline{K_{S}(0, ∞)} \ \{0\}. Moreover u^{∗} > 0, and ∇u^{∗} ≠ 0 because of (2.17).

In order to prove that there exists β > 0 such that

u^{∗}(r, σ) = r^{−β}u^{∗}(1, σ),  (2.18)

we define

Σ_{R} = sup \{C > 0 : Cu^{∗}(x) ≤ u^{∗}(Rx), ∀x ∈ \overline{K_{S}(0, ∞)} \ \{0\}\} .

Note that Σ_{R} exists because of (2.17). If we assume now that the equality

Σ_{R}u^{∗}(x) = u^{∗}(Rx),  (2.19)

does not hold in \overline{K_{S}(0, ∞)}, then

Σ_{R}u^{∗}(x) < u^{∗}(Rx),  (2.20)

from the strong maximum principle and Hopf lemma. Thus the function

θ(ρ) = min_{|x|=ρ} u^{∗}(Rx)/u^{∗}(x),

is strictly monotone and either (i) lim_{ρ→∞} θ(ρ) = Σ_{R}, or (ii) lim_{ρ→0} θ(ρ) = Σ_{R}.

The treatment of the two cases is similar, then we assume (i). For any ρ, there exists σ_{ρ} ∈ S such that

θ(ρ) = u^{∗}(Rρσ_{ρ})/u^{∗}(ρσ_{ρ}).

We can extract a sequence {R_{n_{k}}} such that lim_{n_{k}→∞} R_{n_{k}}/R_{n_{k+1}} = 0. Thus we set

ρ_{n_{k}} = R_{n_{k}}/R_{n_{k+1}} and assume that σ_{ρ_{n_{k}}} → σ_{0} ∈ S̄, by compactness. Because

lim_{n_{k}→∞} θ(ρ_{n_{k}}) = lim_{n_{k}→∞} \frac{C(R_{n_{k+1}})u(R_{n_{k+1}}Rσ_{n_{k}})}{C(R_{n_{k+1}}R)u(R_{n_{k+1}}σ_{n_{k}})},

it implies

Σ_{R} = u^{∗}(R, σ_{0}) < u^{∗}(1, σ_{0}),  (2.21)

which contradicts (2.20).

The last point is to prove that

Σ_{R} = R^{−β}  (2.22)

for some β > 0. Clearly R ↦ Σ_{R} is C^{1} (as u^{∗}) and decreases. For k ∈ ℕ_{∗} there holds

Σ_{R^{k}}u^{∗}(x) = u^{∗}(R^{k}x) = (Σ_{R})^{k}u^{∗}(x).

Then Σ_{R^{k}} = (Σ_{R})^{k}. Consequently, for any m ∈ ℕ_{∗}, Σ_{R^{k/m}} = (Σ_{R})^{k/m}, and finally

Σ_{R^{α}} = (Σ_{R})^{α},

for any positive α. A straightforward consequence is that (2.22) holds for some β > 0. If we set

ω(σ) = u^{∗}(1, σ),  (2.23)

then ω satisfies (2.8) in S, where it is positive, and vanishes on ∂S.

Uniqueness of the couple (β, ω) with sup_{S} ω = 1 follows from the equivalence principle.

Remark Although the extension is far from being obvious, the regularity requirement on the domain S can be relaxed. It is possible to replace it by the assumption that ∂S is piecewise smooth. In dimension 3, Hopf lemma at a corner is replaced by an expansion in terms of conical functions as in Theorem 2.6. In higher dimension the proof goes by induction. However, uniqueness of the couple (β, ω) is not clear. From this observation, we can construct p-harmonic functions in ℝ^{N} \ {0} under the form (2.7) with a finite symmetry group G generated by reflections through hyperplanes. Taking S to be a fundamental simplicial domain of G, we construct (β, ω) in S and then extend ω to the whole sphere by reflections through the edges.

It is natural to imbed this problem in a more general setting, by replacing (S^{N−1}, g_{0}) by a compact and complete d-dimensional Riemannian manifold (M, g). Let ∇_{g}. and ∇_{g} be respectively the divergence operator acting on vector fields on M and the gradient operator. For β ∈ ℝ consider the equation

− ∇_{g}. ((β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2}∇_{g}ψ)

= β((β + 1)(p − 1) − d)(β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2}ψ. (2.24)

Definition We denote by 𝔖_{p}(M) the set of couples (β, ψ) ∈ ℝ × C^{1}(M) satisfying (2.24) and call it the p-quasi-spectrum of M.

Theorem 2.7 If (β, ψ) ∈ 𝔖_{p}(M), then either β((β + 1)(p − 1) − d) = 0 and ψ is any constant, or β((β + 1)(p − 1) − d) > 0 and

∫_{M}(β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2}ψdv_{g} = 0.  (2.25)

Proof From (2.24),

β((β + 1)(p − 1) − d) ∫_{M}(β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2}ψdv_{g} = 0.  (2.26)

Thus if the integral term is not zero β((β + 1)(p − 1) − d) = 0. Clearly if β = 0, ψ is a constant. If β ≠ 0, (β + 1)(p − 1) = d and from (2.24) there holds

−∇_{g}. ((β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2}∇_{g}ψ) = 0,

which implies

∫_{M} (β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2} |∇_{g}ψ|^{2}dv_{g} = 0.

Thus ψ is constant. Moreover if β((β + 1)(p − 1) − d) = 0 any constant satisfies (2.24). Assume now that β((β +1)(p−1)−d) ≠ 0. Then (2.25) holds. Moreover

∫_{M} (β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2} |∇_{g}ψ|^{2}dv_{g}

= β((β + 1)(p − 1) − d) ∫_{M}(β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(p−2)/2}ψ^{2}dv_{g}, (2.27)

and the inequality β((β + 1)(p − 1) − d) > 0 follows.

Remark It should be interesting to study the links between 𝔖_{p}(M) and the geometry of M, in particular the infimum of the β((β + 1)(p − 1 − d). Since we conjectured that the set of such β is unbounded, as on the sphere, their asymptotic distribution could be of interest. In the particular case where p = d + 1, the (d + 1)-quasi-spectrum of M is the set of couples (β, ψ) such that ψ is a solution of

−∇_{g}. ((β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(d−1)/2}∇_{g}ψ) = dβ^{2}(β^{2}ψ^{2} + |∇_{g}ψ|^{2})^{(d−1)/2}ψ. (2.28)

As in the case p = 2, it should be interesting to study the invariance properties of 𝔖_{d+1}(M) with respect to the conformal transformations of M.

3 Equations with strong absorption

In this section we assume N ≥ p > 1 and q > p − 1. If we look for solutions u of (1.5) with ε = 1 under the form (2.7) then β = p/(q + 1 − p) = β_{q} and ω solves

−∇_{σ}. ((β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}∇_{σ}ω) + |ω|^{q−1}ω = λ_{q}(β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}ω,

(3.1)

in S^{N−1}, where

λ_{q} = β_{q}((β_{q} + 1)(p − 1) + 1 − N) = (\frac{p}{q + 1 − p})(\frac{pq}{q + 1 − p} − N). (3.2)

Since

∫_{S^{N−1}} ((β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2} (|∇_{σ}ω|^{2} − λ_{q}ω^{2}) + |ω|^{q+1}) dσ = 0,

there is no solution if λ_{q} ≤ 0 or equivalently if q ≥ N(p − 1)/(N − p). This fact corresponds to a removability result which was proved by Vazquez and Véron [17].

Theorem 3.1 Let Ω be an open subset of ℝ^{N}containing 0, Ω^{∗} = Ω \ {0}, N > p > 1, q ≥ N(p − 1)/(N − p) = p^{#} and g a continuous real valued function satisfying

lim inf_{r→∞} r^{−p^{#}}g(r) > 0, and lim sup_{r→−∞} |r|^{−p^{#}}g(r) < 0.  (3.3)

If u ∈ C(Ω^{∗}) ∩ W_{loc}^{1,p}(Ω^{∗}) is a weak solution of

−∇. (|∇u|^{p−2}∇u) + g(u) = 0, in Ω^{∗},  (3.4)

it can be extended to Ω as a continuous solution of the same equation in whole Ω.

On the contrary, if p − 1 < q < p^{#}, the function

x ↦ u_{s}(x) = γ_{N,p,q}|x|^{−β_{q}},  (3.5)

with

γ_{N,p,q} = ((\frac{p}{q + 1 − p})^{p−1}(\frac{pq}{q + 1 − p} − N))^{1/(q+1−p)},  (3.6)

is a singular solution of

−∇.(|∇u|^{p−2}∇u) + |u|^{q−1}u = 0.  (3.7)

in ℝ^{N} \ {0}. Friedman and Véron provided in [5] a full classification of singular nonnegative solutions of this equation.

Theorem 3.2 Let Ω be an open subset of ℝ^{N}containing 0, Ω^{∗} = Ω \ {0}, N ≥ p > 1, and p − 1 < q < p^{#}, p − 1 < q if p = N. If u ∈ C^{1}(Ω^{∗}) is a nonnegative solution of (3.7) in Ω^{∗}, the following dichotomy occurs.

(i) Either lim_{x→0} |x|^{β_{q}}u(x) = γ_{N,p,q}.

(ii) Either there exists γ > 0 such that lim_{x→0}u(x)/μ_{p}(x) = γ, and u satisfies

−∇.(|∇u|^{p−2}∇u) + |u|^{q−1}u = c_{N,p}|γ|^{p−2}γδ_{0}, in D′(Ω).  (3.8)

(iii) Or u can be extended to whole Ω as a C^{1} solution of (3.7) in Ω.

Proof By scaling we can always assume that B_{1} ⊂ Ω. The starting point is an a priori estimate of Keller-Osserman type due to Vazquez [16]: if u is any solution of (3.7) in B_{1}^{∗} = {x ∈ ℝ^{N} : 0 < |x| < 1}, there exists a positive constant K = K_{N,p,q} such that

|u(x)| ≤ K|x|^{−β_{q}},  (3.9)

for any 0 < |x| ≤ 1/2. By writting (3.7) under the form

−∇.(|∇u|^{p−2}∇u) + d(x)u^{p−1} = 0,

with d(x) = u^{q+1−p}, and using the Trudinger’s estimate [15] in Harnack inequality, it follows that there exists some A = A(N, p, q) > 0 such that

max_{|x|=r} u(x) ≤ A min_{|x|=r} u(x),

for any 0 < r ≤ 1/4.

Step 1 Assume that u(x)/μ_{p}(x) is not bounded in a neighborhood of 0. The previous estimate implies that there exists a sequence r_{n} → 0 such that

lim_{r_{n}→0} min_{|x|=r_{n}} u(x)/μ_{p}(r_{n}) = ∞.

Consequently, for any k > 0 there exists some n_{k} such that for n ≥ n_{k} the function u is bounded from below in B̄_{1} \B_{r_{n}} by the solution v_{n} of the Dirichlet problem

\begin{matrix}−∇.(|∇v_{n}|^{p−2}∇v_{n}) + |v|_{n}^{q−1}v_{n} = 0, in B_{1} \ B̄_{r_{n}}, \\ v_{n}(x) = 0 if |x| = 1, \\ v_{n}(x) = kμ_{p}(r_{n}) if |x| = r_{n}. \end{matrix}  (3.10)

Note that v_{n} is positive, radial and bounded from above by kμ_{p}(x). Since q < p^{#} the absorption term v_{n}^{q} satisfies

∫_{r_{n}}^{1} v_{n}^{q}r^{N−1}dr ≤ k^{q} ∫_{0}^{1} μ_{p}^{q}(r)^{q}r^{N−1}dr,

independently of n. This is sufficient to derive that there exists

lim_{r_{n}→0} v_{n} = v,

where v = v_{(k)} is a radial solution of

\begin{matrix}−∇.(|∇v|^{p−2}∇v) + |v|^{q−1}v = 0, in B_{1} \ \{0\}, \\ v(x) = 0 if |x| = 1, \\ v(x) ≈ kμ_{p}(x) if |x| → 0. \end{matrix}  (3.11)

Actually, v is nonnegative, radial, bounded from above by u and solves

−∇.(|∇v|^{p−2}∇v) + v^{q} = c_{N,p}k^{p−1}δ_{0}, in D′(B_{1}).  (3.12)

When k → ∞, v_{(k)} increases and converges to some v_{(∞)} which is a positive and radial solution of (3.7) in B_{1}^{∗} such that

lim_{r→0} v_{(∞)}(r)/μ_{p}(r) = ∞.  (3.13)

Moreover

v_{(∞)}(|x|) ≤ u(x) ≤ u_{s}(x) = γ_{N,p,q}|x|^{−β_{q}} in B_{1}^{∗}.  (3.14)

The analysis of the behavior of v_{(∞)} near r = 0 is done either by a technical O.D.E. analysis, or a scaling invariance method based on uniqueness of the radial solution of (3.11) (see [4] for a proof in the case p = 2). From this analysis follows

lim_{r→0} r^{β_{q}}v_{(∞)}(r) = γ_{N,p,q}.  (3.15)

Consequently

lim_{x→0} |x|^{β_{q}}u(x) = γ_{N,p,q}.  (3.16)

Step 2 Assume that u(x)/μ_{p}(x) is bounded near 0 (in this case, we need not impose the positivity of u). In such a case the absorption term |u|^{q−1}u is dominated by Cμ_{p}^{q} for some C > 0. By using the same scaling methods, estimates on ∇u, and the strict comparison principle as in the proof of Theorem 2.1, it can be proved that there exists a real number γ such that

lim_{x→0} u(x)/μ_{p}(x) = γ,  (3.17)

and

lim_{x→0}(|x|)^{(N−1)/(p−1)}∇ (u(x) − γμ_{p}(x)) = 0.  (3.18)

Thus u satisfies (3.8). If γ = 0, then

|u(x)| ≤ max_{|y|=1} |u(y)|, ∀x ∈ B_{1},

by the maximum principle. Thus u is C^{1,α} by the regularity theory of quasilinear equations. □

The construction of nodal singular solutions of (3.7) under the form (2.7) is done by a shooting technique, as for the p-Laplace equation.

Theorem 3.3 Let 0 < p − 1 < q < p^{#} and S ⊂ S^{N−1} be a domain with a C^{2} relative boundary ∂S. Let β = β_{S} > 0 be the exponent defined in Theorem 2.6. If β_{q} > β_{S} there exists a positive solution ω of (3.1) in S which vanishes on ∂S.

Proof: Step 1 Construction of an approximate solution. For ε > 0 small enough denote by u = u_{ε} the unique solution of

−∇. (|∇u|^{p−2}∇u) + |u|^{q−1}u = 0, in K_{S}(1, ∞),

u = εg^{β_{q}}, on ∂K_{S}(1, ∞),

lim sup_{|x|→∞} |x|^{β_{q}}u(x) < ∞.

(3.19)

By the monotone operator theory, u is unique and satisfies 0 ≤ u < u_{s}.

Step 2 Construction of a minorant subsolution. Let ω = ω_{S} be the corresponding second element of the couple (β, ω) = (β_{S}, ω_{S}) obtained in Theorem 2.6. Put θ = β_{q}/β_{S}. We claim that for δ > 0 small enough, the function

(r, σ) ↦ w_{δ}(x) = w_{δ}(r, σ) = r^{−β_{q}}δω_{S}^{θ}(σ)  (3.20)

satisfies

\begin{matrix}−∇. (|∇w_{δ}|^{p−2}∇w_{δ}) + |w_{δ}|^{q−1}w_{δ} ≤ 0, in K_{S}(1, ∞), \\ w_{δ} = 0, on B_{S}(1, ∞). \end{matrix}  (3.21)

Set

Lw_{δ} = −∇. (|∇w_{δ}|^{p−2}∇w_{δ}) + |w_{δ}|^{q−1}w_{δ}.

Then L(w_{δ}) = r^{−qβ_{q}}T (δω_{S}^{θ}), where

T (η) = −∇_{σ}. ((β_{q}^{2}η^{2} + |∇_{σ}η|^{2})^{(p−2)/2}∇_{σ}η)−λ_{q}(β_{q}^{2}η^{2}+|∇_{σ}η|^{2})^{(p−2)/2}η+|η|^{q−1}η.

Putting η = δω_{S}^{θ},

(β_{q}^{2}η^{2} + |∇_{σ}η|^{2})^{(p−2)/2} = δ^{p−2}θ^{p−2}ω_{S}^{(θ−1)(p−2)}(β_{S}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2},

and

∇_{σ}.((β_{q}^{2}η^{2} + |∇_{σ}η|^{2})^{(p−2)/2}∇_{σ}η)
=δ^{p−1}θ^{p−1}∇_{σ}.(ω_{S}^{(θ−1)(p−1)}(β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{(p−2)/2}∇_{σ}ω_{S})
=δ^{p−1}θ^{p−1}ω_{S}^{(θ−1)(p−1)}∇_{σ}.((β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{(p−2)/2}∇_{σ}ω_{S})

+ (θ − 1)(p − 1)δ^{p−1}θ^{p−1}ω_{S}^{(θ−1)(p−1)−1}(β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{(p−2)/2}|∇_{σ}ω_{S}|^{2}

But

−∇_{σ}. ((β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{(p−2)/2}∇_{σ}ω_{S}) = λ_{S}(β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{(p−2)/2}ω_{S}, with λ_{S} = (β_{S} + 1)(p − 1) + 1 − N). Thus,

δ^{1−p}T (η) = δ^{q+1−p}ω_{S}^{θq} + ω_{S}^{(θ−1)(p−1)−1}θ^{p−2}(β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{(p−2)/2}
× ((θλ_{S} − λ_{q})ω_{S}^{2} − θ(θ − 1)(p − 1)|∇_{σ}ω_{S}|^{2}) .

Since θλ_{S} − λ_{q} = β_{q}(β_{S} − β_{q})(p − 1) = −β_{S}^{2}θ(θ − 1)(p − 1),

δ^{1−p}T (η)

= δ^{q+1−p}ω_{S}^{θq} − (p − 1)(θ − 1)θ^{p−1}ω_{S}^{(θ−1)(p−1)−1}(β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{p/2}
≤ δ^{q+1−p}ω_{S}^{θq} − (p − 1)(θ − 1)θ^{p−1}ω_{S}^{θ(p−1)}.

by assumption θ > 1, therefore there exists δ > 0 such that T (η) ≤ 0. Moreover it can also be assumed that δω_{S}^{θ} ≤ ε. Then w_{δ}(x) ≤ u(x) if |x| = 1 and w_{δ} ≤ u in K_{S}(1, ∞) by the maximum principle. Henceforth

δω_{S}^{θ}(x/|x|) ≤ |x|^{β_{q}}u(x) ≤ γ_{N,p,q} in K_{S}(1, ∞).  (3.22)

Step 3 For R > 0, define the function u_{R} by u_{R} = R^{β_{q}}u(Rx). The function u_{R} satisfies (3.7) in K_{S}(1/R, ∞). By the degenerate elliptic equation regularity theory, the set of functions {u_{R}} remains bounded in the C_{loc}^{1,α}-topology of \overline{K_{S}(0, ∞)}\\{0\}. Let 0 < R < R′, in order to compare u_{R} and u_{R′} in K_{S}(1/R, ∞) we recall that g(x) = (2 − |x|)_{+}. The relation

R^{′β_{q}}(2 − R′|x|)_{+}^{β_{q}} ≤ R^{β_{q}}(2 − R|x|)_{+}^{β_{q}} for |x| ≥ 1/R,

implies

\frac{d}{dR} (R^{β_{q}}(2 − R|x|)_{+}^{β_{q}}) ≤ 0 for |x| ≥ 1/R,

If and only if

β_{q}R(2 − R|x|)_{+}^{β_{q}−1}(2 − 2R|x|) ≤ 0 for |x| ≥ 1/R,

which holds true. By the maximum pinciple

R′ ≥ R ⟹ u_{R′} ≤ u_{R} ∈ K_{S}(1/R, ∞).  (3.23)

Thus there exists a function u^{∗} such that u_{R} decreases and converges to u^{∗} as R → ∞ in C_{loc}^{1}(\overline{K_{S}(0, ∞)} \ \{0\}). The function u^{∗} is a solution of (3.7) in K_{S}(0, ∞) which vanishes on B_{S}(0, ∞). Because of (3.22), u^{∗} satisfies

δω_{S}^{θ}(x/|x|) ≤ |x|^{β_{q}}u^{∗}(x) ≤ γ_{N,p,q} in K_{S}(0, ∞).  (3.24)

Finally,

lim_{R→∞} R^{β_{q}}u(Rr, σ) = u^{∗}(r, σ) = r^{−β_{q}} lim_{R→∞}(Rr)^{β_{q}}u(Rr, σ) = r^{−β_{q}}u^{∗}(1, σ).

Putting ω = u^{∗}(1, σ) completes the proof. □ In the next theorem we prove that the condition β_{q} > β_{S} is sharp.

Theorem 3.4 Let 0 < p − 1 < q < p^{#} and S ⊂ S^{N−1} be a domain with a C^{2} relative boundary ∂S. If β_{q} ≤ β_{S} there exists no solution ω of (3.1) in S which vanishes on ∂S.

Proof Assume ω is a solution of (3.1). If θ = β_{q}/β_{S}, then 0 < θ ≤ 1. If we denote again η = δω_{S}^{θ}, for some δ > 0, it follows from the proof of Theorem 3.3-Step 2 that, for any δ > 0,

δ^{1−p}T (η) = δ^{q+1−p}ω_{S}^{θq}

+(p − 1)(1 − θ)θ^{p−1}ω_{S}^{(θ−1)(p−1)−1}(β_{S}^{2}ω_{S}^{2} + |∇_{σ}ω_{S}|^{2})^{p/2} > 0.

We take δ = δ_{0} as the smallest parameter such that η = η_{δ} ≥ ω. Notice that such a choice is always possible since ω ∈ C^{1}(S̄), the normal derivative of ω_{S} on the relative boundary ∂S is negative from the Hopf boundary lemma and therefore ω_{S}^{θ}(σ) ≥ c(dist(σ, ∂S)^{θ} for some c > 0. We shall distinguish according there exists σ_{0} ∈ S such that

η(σ) ≥ ω(σ), ∀σ ∈ S̄, and η(σ_{0}) = ω(σ_{0}),  (3.25)

or not. If (3.25) holds true, which is always the case if β_{S} > β_{q}, the function ψ = η − ω is nonnegative in S̄, not identically 0 and achieves its minimal value 0 in an interior point σ_{0}. Let g = (g_{ij}) be the metric tensor on S^{N−1}. We write in local coordinates σ_{j} around σ_{0},

|∇φ|^{2} = ∑_{j,k} g^{jk} \frac{∂φ}{∂σ_{j}} \frac{∂φ}{∂σ_{k}} ,

∇.X = \frac{1}{\sqrt{|g|}} ∑_{ℓ} \frac{∂}{∂σ_{ℓ}} (\sqrt{|g|}X^{ℓ}) = \frac{1}{\sqrt{|g|}} ∑_{ℓ,i} \frac{∂}{∂σ_{ℓ}} (\sqrt{|g|}g^{ℓi}X_{i}) ,

if we lower the indices by setting X^{ℓ} = ∑_{i} g^{ℓi}X_{i}. From the Mean Value Theo-

rem, we obtain

(β_{q}^{2}η^{2} + |∇_{σ}η|^{2})^{(p−2)/2} \frac{∂η}{∂σ_{i}} − (β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2} \frac{∂ω}{∂σ_{i}}

= ∑_{j} α_{j}^{i} \frac{∂(η − ω)}{∂σ_{j}} + b^{i}(η − ω),

where

b^{i} = (p − 2) (β_{q}^{2}(ω + t(η − ω))^{2} + |∇_{σ}(ω + t(η − ω))|^{2})^{(p−4)/2}

×(ω + t(η − ω))\frac{∂(ω + t(η − ω))}{∂σ_{i}},

and

α_{j}^{i} = (p − 2) (β_{q}^{2}(ω + t(η − ω))^{2} + |∇_{σ}(ω + t(η − ω))|^{2})^{(p−4)/2}

×\frac{∂(ω + t(η − ω))}{∂σ_{i}} ∑_{k} g^{jk} \frac{∂(ω + t(η − ω))}{∂σ_{k}}

+δ_{i}^{j} (β_{q}^{2}(ω + t(η − ω))^{2} + |∇_{σ}(ω + t(η − ω))|^{2})^{(p−2)/2} .

Since the graph of η and ω are tangent at σ_{0},

η(σ_{0}) = ω(σ_{0}) = P_{0} > 0 and ∇η(σ_{0}) = ∇ω(σ_{0}) = Q.

Thus

b^{i}(σ_{0}) = (p − 2) (β_{q}^{2}P_{0}^{2} + |Q|^{2})^{(p−4)/2} P_{0}Q_{i},

and

α_{j}^{i}(σ_{0}) = (β_{q}^{2}P_{0}^{2} + |Q|^{2})^{(p−4)/2}(δ_{i}^{j}(β_{q}^{2}P_{0}^{2} + |Q|^{2}) + (p − 2)Q_{i} ∑_{k} g^{jk}Q_{k}).

Now

T (η) − T (ω)
=\frac{−1}{\sqrt{|g|}} ∑_{ℓ,i} \frac{∂}{∂σ_{ℓ}} [\sqrt{|g|}g^{ℓi}((β_{q}^{2}η^{2} + |∇_{σ}η|^{2})^{\frac{p}{2} −1} \frac{∂η}{∂σ_{i}} − (β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{\frac{p}{2} −1} \frac{∂ω}{∂σ_{i}} )]

− λ_{q} ((β_{q}^{2}η^{2} + |∇_{σ}η|^{2})^{\frac{p}{2} −1}η − (β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{\frac{p}{2} −1}ω) + η^{q} − |ω|^{q−1}ω),

= − \frac{1}{\sqrt{|g|}} ∑_{ℓ,i} \frac{∂}{∂σ_{ℓ}} [\sqrt{|g|}g^{ℓi}( ∑_{j} α_{j}^{i} \frac{∂(η − ω)}{∂σ_{j}} + b^{i}(η − ω))]

+ ∑_{i} C_{i} \frac{∂(η − ω)}{∂σ_{i}} + C(η − ω)

= − \frac{1}{\sqrt{|g|}} ∑_{ℓ,j} \frac{∂}{∂σ_{ℓ}} [a_{j}^{ℓ} \frac{∂(η − ω)}{∂σ_{j}}] + ∑_{i} C_{i} \frac{∂(η − ω)}{∂σ_{i}} + C(η − ω),

where the C_{i} and C are continuous functions and

a_{j}^{ℓ} = \sqrt{|g|} ∑_{i} g^{ℓi}α_{j}^{i}.

The matrix (α_{j}^{i}(σ_{0})) is symmetric, definite and positive since it is the Hessian of the strictly convex function

X = (X_{1}, …, X_{n−1}) ↦ \frac{1}{p} (P_{0}^{2} + |X|^{2})^{p/2} = \frac{1}{p}(P_{0}^{2} + ∑_{j,k} g^{jk}X_{j}X_{k})^{p/2}.

Therefore, (α_{j}^{i}) has the same property in some neighborhood of σ_{0}, and the same holds true with (a_{j}^{ℓ}). Finally the function ψ = η − ω is nonnegative, vanishes at σ_{0} and satisfies

−\frac{1}{\sqrt{|g|}} ∑_{ℓ,j} \frac{∂}{∂σ_{ℓ}} [a_{j}^{ℓ} \frac{∂ψ}{∂σ_{j}} ] + ∑_{i} C_{i} \frac{∂ψ}{∂σ_{i}} + C_{+}ψ ≥ 0.  (3.26)

Then ψ = 0 in a neighborhood of S. Since S is connected, ψ is identically 0, which a contradiction.

If (3.25) does not hold, then θ = 1 and that the graphs of η and ω are tangent at some point σ_{0} of the relative boundary ∂S. Proceeding as above and using the fact that ∂η/∂ν exists and never vanishes on the boundary, we see that ψ = η − ω satisfies (3.26) with a strongly elliptic operator in a neighborhood N of σ_{0}. Moreover ψ > 0 in N, ψ(σ_{0}) = 0 and ∂ψ/∂ν(σ_{0}) = 0. This is a contradiction, which ends the proof.

Remark The existence result of Theorem 3.3 is valid if S is no longer a C^{2} domain but a domain with a piecewise regular boundary since only the existence of (β_{S}, ω_{S}) is needed. We conjecture that the condition β_{q} > β_{S} is still necessary. As is section 2, we can construct nodal solutions of (3.1) with a finite symmetry group G generated by reflections through hyperplanes. Taking S to be a fundamental simplicial domain of G, we construct (β, ω) in S and then extend ω to the whole sphere by reflections through the edges. It follows that there exists nodal singular solutions of (3.7) in ℝ^{N} \ {0}.

Remark Under the assumptions of Theorem 3.3, we conjecture that uniqueness of the positive solution ω of (3.1) which vanishes on ∂S holds. If S = S^{N−1} and p − 1 < q < p^{#}, an application of the maximum principle (or a consequence of Theorem 3.2) implies that the only positive solution of (3.1) on S^{N−1} is the constant function γ_{N,p,q}.

4 Equations with a source term

If we look for solutions of

∇.(|∇u|^{p−2}∇u) + |u|^{q−1}u = 0  (4.1)

under the form (2.7), then β = p/(q + 1 − p) = β_{q} and ω solves

∇_{σ}. ((β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}∇_{σ}ω) + |ω|^{q−1}ω + λ_{q}(β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}ω = 0,

(4.2)

on S^{N−1} with λ_{q} defined by (3.2). By integrating (4.2) we get

λ_{q} ∫_{S^{N−1}}(β_{q}^{2}ω^{2} + |∇_{σ}ω|^{2})^{(p−2)/2}ωdσ + ∫_{S^{N−1}} |ω|^{q−1}ωdσ = 0.

Therefore, there exists no positive solution if λ_{q} ≥ 0, or equivalently q ≤ N(p − 1)/(N − p) (it is always assumed that q > p − 1). In the range 1 < p < N and q > N(p − 1)/(N − p) the constant function

ω_{0} = (β_{q}^{p−1}(N − qβ_{q})^{1/(q+1−p)}

is a solution of (4.2), and a natural question is to look for nonconstant solutions. As in Section 2, we imbed this problem in the more general setting of a compact d-dimensional Riemannian manifold (M, g) without boundary. For β and λ ∈ ℝ consider the equation

−∇_{g}. ((β^{2}ω^{2} + |∇_{g}ω|^{2})^{(p−2)/2}∇_{g}ω) + λ(β^{2}ω^{2} + |∇_{g}ω|^{2})^{(p−2)/2}ω = |ω|^{q−1}ω.

(4.3)

We shall assume λ > 0 in order for the constant solution

ω_{∗} = (β^{p−2}λ)^{1/(q+1−p)}

to exist. We assume also that the starting equation is super-quasilinear in the sense that β > 0 and q > q + 1 − p. We can linearize (4.3) in a neighborhood of ω_{∗}, and we obtain

\begin{matrix}\frac{d}{dt}∇_{g}. ((β^{2}(ω_{∗} + tφ)^{2} + |∇_{g}(ω_{∗} + tφ)|^{2})^{(p−2)/2}∇_{g}(ω_{∗} + tφ)) & |_{t=0} \\ = β^{p−2}ω_{∗}^{p−2}Δ_{g}φ. \end{matrix}

\frac{d}{dt} ((β^{2}(ω_{∗} + tφ)^{2} + |∇_{g}(ω_{∗} + tφ)|^{2})^{(p−2)/2}(ω_{∗} + tφ)) |_{t=0} = (p − 1)β^{p−2}ω_{∗}^{p−2}φ.

\frac{d}{dt} (ω_{∗} + tφ)^{q} |_{t=0} = qω_{∗}^{q−1}φ.

Since ω_{∗} = (β^{p−2}λ)^{1/(q+1−p)}, the linearized equation is

−Δ_{g}φ = (q + 1 − p)λφ.  (4.4)

where Δ_{g} = ∇_{i}∇^{i} is the laplacian on M.

Theorem 4.1 Let μ_{1} be the first nonzero eigenvalue of Δ_{g}, and assume it is simple. Then for any λ > μ_{1}/(q + 1 − p) equation (4.3) admits a nonconstant positive solution ω_{λ}.

Proof The existence of a global and unbounded branch of bifurcation B = {(λ, ω_{λ})} ⊂ ℝ × C^{1}(M) issued from (μ_{1}/(q + 1 − p), ω_{∗}) follows from the application in the space C^{1}(M) of the classical bifurcation theorem from a simple eigenvalue. □

Remark The condition on the simplicity of μ_{1} can be avoided in many cases where symmetries occur. When (M, g) = (S^{N−1}, g_{0}), we have the parametric representation

S^{N−1} = {σ = (cos φ, sin φσ′) : φ ∈ [0, π], σ′ ∈ S^{N−2}},

and

Δ_{S^{N−1}}ω = sin^{2−N} φ\frac{∂}{∂φ}( sin^{N−2} φ\frac{∂ω}{∂φ}) + sin^{−2} φΔ_{S^{N−2}}ω.

If we only consider function depending on φ (they are called zonal functions), μ_{1} = N − 1 is a simple eigenvalue. Moreover any eigenspace of S^{N−1} contains a 1-dimensional sub-eigenspace of functions depending only on φ. Therefore all the corresponding eigenvalues are simple. Thus from each of the couples (μ_{k}/(q + 1 − p), ω_{∗}) is issued a C^{1} curve of positive solutions (λ, ω_{λ}) with λ > μ_{k}/(q + 1 − p).

Open question An interesting problem is to find sufficient conditions besides λ ≤ μ_{1}/(q + 1 − p) and probably q ≤ dp/(d − p) − 1, in order the constant ω_{∗} be the only positive solution of (4.3). We believe additional conditions linked to the curvature should be found (see [6], [2], [10] in the case p = 2).

We define the critical Sobolev exponent q_{c} by

q_{c} = \frac{Np}{N − p} − 1 = \frac{N(p − 1) + p}{N − p}.  (4.5)

A particular case of equation (4.1) is when q = q_{c}. Then

q_{c} + 1 − p = \frac{p^{2}}{N − p}, β_{q_{c}} = \frac{N − p}{p} and λ_{q_{c}} = −β_{q_{c}}^{2}.

The critical equation is therefore

∇_{σ}.((β_{q_{c}}^{2}ω^{2} + |∇_{σ}ω|^{2})^{p/2−1}∇_{σ}ω) + |ω|^{q_{c}−1}ω − β_{q_{c}}^{2}(β_{q_{c}}^{2}ω^{2} + |∇_{σ}ω|^{2})^{p/2−1}ω = 0,

(4.6)

on S^{N−1}. A natural question is to explore the connection between the positive solutions of (4.6) and the positive solutions of

−∇.(|∇u|^{p−2}∇u)v = v^{q_{c}} in ℝ^{N}.  (4.7)

Notice that the radial solutions of this equation, depending of a parameter a > 0, are known:

v_{a}(x) = (Na(\frac{N − p}{p − 1})^{p−1})^{(N−p)/p^{2}} (a + |x|^{p/(p−1)})^{(p−N)/p} .  (4.8)

The solutions of (4.6) are the critical points of the functional

J_{q_{c}}(ψ) = ∫_{S^{N−1}} (\frac{1}{p}(β_{q_{c}}^{2}ψ^{2} + |∇_{σ}ψ|^{2})^{p/2} − \frac{1}{q_{c} + 1}|ψ|^{q_{c}+1}) dσ,  (4.9)

where ψ ∈ W^{1,p}(S^{N−1}).

Remark Let 0 < p − 1 < q < q_{c} and S ⊂ S^{N−1}, it would be interesting to construct positive solutions ω of (4.2) in S which vanish on ∂S. In the case p = 2, the equation becomes

\begin{matrix}−Δ_{σ}ω = β_{q}(β_{q} + 2 − N)ω + ω^{q}, in S, \\ ω = 0, on ∂S, \end{matrix}  (4.10)

where Δ_{σ} is the Laplace-Beltrami operator on the sphere and β_{q} = 2/(q − 1). The solutions are constructed by a standard minimization process with a constraint. If 1 < q < (N + 1)/(N − 3), a necessary and sufficient condition for the existence of such a solution is

β_{q} < β_{S},

and in that case β_{S} = λ_{1}(S) is the first eigenvalue of Δ_{σ} in W_{0}^{1,2}(S). When p ≠ 2, this method no longer works. However under the same condition

β_{q} < β_{S} and q < q_{c},

(adapted to the case of a general p) we have been able to prove the existence of positive super and subsolutions to equation (4.2). Unfortunately we do not know if they are ordered. We conjecture that, in the subcritical case, the condition β_{q} ≥ β_{S} is a necessary and sufficient condition for the existence of positive solutions to (4.2).

We want to mention another quasilinear equation of Emden type which admits specific solutions:

−∇.(|∇u|^{p−2}∇u) = λe^{u},  (4.11)

with λ > 0. If we look for particular solutions of (4.11) under the form

u(r, σ) = α ln r + bw(σ) + k,

where α, b and k are constants, one finds α = −p and

b∇_{σ}.([p^{2} + b^{2}|∇_{σ}w|^{2}]^{p/2−1}∇_{σ}w) + λe^{k}e^{bw} − p(N − p)[p^{2} + b^{2}|∇_{σ}w|^{2}]^{p/2−1} = 0

on S^{N−1}. A necessary condition for the existence of a solution is

p − N < 0.  (4.12)

Assuming this condition, we take b = p and get

∇_{σ}.([1 + |∇_{σ}w|^{2}]^{p/2−1}∇_{σ}w) − (N − p)[1 + |∇_{σ}w|^{2}]^{p/2−1} + λp^{1−p}e^{k}e^{pw} = 0.

Now choose k = ln(p^{p−1}λ^{−1}). Assuming 1 < p < N, then w satisfies

∇_{σ}.([1 + |∇_{σ}w|^{2}]^{p/2−1}∇_{σ}w) − (N − p)[1 + |∇_{σ}w|^{2}]^{p/2−1} + e^{pw} = 0 (4.13)

on S^{N−1}. In the particular case p = 2, N = 3, this is the equation of conformal change of structures on S^{2}, and the set of all solutions can be endowed with a structure of a 3-dim non-compact Lie group. We believe that the case p = N − 1 = n should play a similar algebraic role. The corresponding equation is

∇_{σ}.([1 + |∇_{σ}w|^{2}]^{n/2−1}∇_{σ}w) − [1 + |∇_{σ}w|^{2}]^{n/2−1} + e^{nw} = 0  (4.14)

on S^{N−1}.

In the case 1 < p < N and p − 1 < q < N(p − 1)/(N − p) = p^{#}, the classification of isolated singularities of positive solutions of (4.1) has been initiated by Guedda and Véron [7], under the priori bound assumption (4.18), and then completed by Bidaut-Véron [1].

Theorem 4.2 Let Ω be an open subset of ℝ^{N} containing 0, Ω^{∗} = Ω \ {0}, 1 < p < N and p − 1 < q < p^{#}, and let u ∈ C^{1}(Ω^{∗}) be a nonnegative solution of (4.1) in Ω^{∗}. Then the following dichotomy occurs.

(i) Either there exists α > 0 such that lim_{x→0}u(x)/μ_{p}(x) = α, and u satisfies

−∇.(|∇u|^{p−2}∇u) − u^{q} = c_{N,p}α^{p−1}δ_{0}, in D′(Ω).  (4.15)

(ii) Or u can be extended as a C^{1} solution of (4.1) in Ω.

The general proof of this result is based upon the extension obtained in [1] of the Brezis-Lions lemma [3] dealing with singular super-harmonic functions.

Lemma 4.3 Let 1 < p < N and u ∈ C(Ω^{∗})∩W_{loc}^{1,p}(Ω^{∗}) with ∇. (|∇u|^{p−2}∇u) ∈

L_{loc}^{1}(Ω^{∗}) is a nonnegative solution of

∇. (|∇u|^{p−2}∇u) ≤ 0,  (4.16)

a.e. in Ω and in the sense of distributions in Ω^{∗}. Then u^{p−1} ∈ M_{loc}^{N/(N−p)}(Ω), |∇u|^{p−1} ∈ M_{loc}^{N/(N−1)}(Ω), and there exists a nonnegative constant β and some g ∈ L_{loc}^{1}(Ω) such that

−∇. (|∇u|^{p−2}∇u) = g + βδ_{0},  (4.17)

in the sense of distributions in Ω.

From this result and using some test functions introduced by Serrin in [11], Harnack inequality and a method due to Benilan, it is possible to derive the key estimate that is satisfied by any positive solution u of (4.1) in this range of values of q : there exists some C > 0 such that

u(x) ≤ Cμ_{p}(x),  (4.18)

holds in a neighborhood of 0. With this estimate, a scaling methods similar to the one used in [5] ends the proof. Actually, in [7], a more general convergence result is proved: if 1 < p ≤ N, p − 1 < q < p^{#} (no condition if p = N) and u ∈ C^{1}(Ω^{∗}) is a signed solution of (4.1) in Ω^{∗} such that

|u(x)| ≤ Cμ_{p}(x),

near 0, then either (i’) there exists α ≠ 0 such that lim_{x→0}u(x)/μ_{p}(x) = α, and u satisfies

−∇.(|∇u|^{p−2}∇u) − |u|^{q−1}u = c_{N,p}|α|^{p−2}αδ_{0}, in D′(Ω).  (4.19)

(ii’) Or u can be extended as a C^{1} solution of (4.1) in Ω.

In the case q ≥ p^{#}, the classification of isolated singularities of radial solutions of (4.1) has been performed by Guedda and Véron [7]. Latter on Guedda and Véron’s results have been extended by Bidaut-Véron [1], with no restriction on q, but always when dealing with radial solutions.

Theorem 4.4 Let p^{#} < p < q_{c}, and let u ∈ C^{1}(B_{1}^{∗}) be a radial solution of (4.1) in B_{1}^{∗}. Then the following occurs.

(i) Either u is a regular solution of (4.1) in B_{1}.

(ii) Either

u(x) ≡ (β_{q}^{p−1}(N − qβ_{q})^{1/(q+1−p)}|x|^{−β_{q}},

or

u(x) ≡ −(β_{q}^{p−1}(N − qβ_{q})^{1/(q+1−p)}|x|^{−β_{q}}.

(iii) Or |x|^{β_{q}}u(x) is not constant and

lim_{x→0} |x|^{β_{q}}u(x) = (β_{q}^{p−1}(N − qβ_{q})^{1/(q+1−p)}|x|^{−β_{q}},

or

lim_{x→0} |x|^{β_{q}}u(x) = −(β_{q}^{p−1}(N − qβ_{q})^{1/(q+1−p)}|x|^{−β_{q}}.

The results related to the cases p^{#} = p, p = Np/(N − p) − 1 and p > Np/(N − p) − 1 can be found in [1]. For a long time, the non-radial case appeared out of reach up to the recent work of Serrin and Zou [13]. In this striking paper they proved, among other results, that Gidas and Spruck classical a priori estimate in the case p = 2, N/(N − 2) ≤ q < q_{c} [6] still holds in the range p > 1 and p^{#} ≤ p < Np/(N − p) − 1 (under a form appropriate to the p-Laplace operator). Any positive solution u of (4.1) in Ω^{∗} satisfies

u(x) ≤ C|x|^{−β_{q}},  (4.20)

near 0. The proof is an extremely clever (but difficult) adaptation of the proof given by Gidas and Spruck. Among other results Serrin and Zou provide also a description of entire solutions of the same equation in ℝ^{N}.

References

[1] Bidaut-Véron M. F., Local and global properties of solutions of quasilinear elliptic equations of Emden-Fowler type, Arch. Rat. Mech. Anal. 107, 293-324 (1989).

[2] Bidaut-Véron M. F. & Véron L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math. 106, 489-539 (1991).

[3] Brezis H. & Lions P. L., A note on isolated singularities for linear elliptic equations, Math. Anal. Appl. Adv. Suppl. Stud 7A, 263-266 (1981).

[4] Brezis H. & Oswald L., Singular solutions for some semilinear elliptic equations, Arch. Rat. Mech. Anal. 99, 249-257 (1987).

[5] Friedman A. & Véron L., Singular solutions of some quasilinear elliptic equations, Arch. Rat. Mech. Anal. 96, 359-387 (1986).

[6] Gidas B. & Spruck J., Local and global behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34, 525-598 (1980).

[7] Guedda M. & Véron L., Local and global properties of solutions of quasilinear elliptic equations, J. Diff. Equ. 76, 159-189 (1988).

[8] Kroll I. N. & Mazja V. G., The lack of continuity and Hölder continuity of the solution of a certain quasilinear equation, Proc. Steklov Inst. Math. 125, 130-136 (1973).

[9] Kichenassamy S. & Véron L., Singular solutions of the p-Laplace equation, Math. Ann. 275, 599-615 (1986).

[10] Licois J. R. & Véron L., A class of nonlinear conservative elliptic equations in cylinder, Ann. Scu. Norm. Sup. Pisa Ser. IV,Vol XXVI, 249-283 (1998).

[11] Serrin J., Local behaviour of solutions of quasilinear equations, Acta Math. 111, 247-302 (1964).

[12] Serrin J., Isolated singularities of solutions of quasilinear equations, Acta Math. 113, 219-240 (1965).

[13] Serrin J. & Zou H., Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. (to appear).

[14] Tolksdorff P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Part. Diff. Equ. 8, 773-817 (1983).

[15] Trudinger N., On Harnack type inequalities and their applications to quasilinear elliptic equations, Comm. Pure Appl. Math. 20, 721-747 (1967).

[16] Vazquez J. L., An a priori interior estimate for the solutions of a nonlinear problem representing weak diffusion, Nonlinear Anal. T., M., & A. 5, 95-103 (1980).

[17] Vazquez J. L. & Véron L., Removable singularities of some strongly nonlinear elliptic equations, Manuscripta Math. 33, 129-144 (1980).

[18] Véron L., Singularities of solutions of second order quasilinear elliptic equations, Pitman Research Notes in Math. 353, Addison-Wesley- Longman (1996).

Laurent Véron Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6063 Faculté des Sciences et Techniques Parc de Grandmont, France e-mail: veronl@univ-tours.fr